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An extension of almost sure central limit theory

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  • Hörmann, Siegfried

Abstract

We show that if Xn are i.i.d. random variables with and (dk) is a positive numerical sequence obeying a condition similar to Kolmogorov's condition for the LIL, then letting we haveThis shows that logarithmic means, used traditionally in a.s. central limit theory, can be replaced by other means lying much closer to ordinary (Cesàro) averages, leading to considerably sharper results. Our results remain valid (under suitable regularity conditions) for independent random variables Xn satisfying the weak limit theorem with an arbitrary distribution function H.

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  • Hörmann, Siegfried, 2006. "An extension of almost sure central limit theory," Statistics & Probability Letters, Elsevier, vol. 76(2), pages 191-202, January.
    • Handle: RePEc:eee:stapro:v:76:y:2006:i:2:p:191-202
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    References listed on IDEAS

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    1. Lacey, Michael T. & Philipp, Walter, 1990. "A note on the almost sure central limit theorem," Statistics & Probability Letters, Elsevier, vol. 9(3), pages 201-205, March.
    2. Berkes, István & Csáki, Endre, 2001. "A universal result in almost sure central limit theory," Stochastic Processes and their Applications, Elsevier, vol. 94(1), pages 105-134, July.
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